L03. PROBABILITY REVIEW II COVARIANCE PROJECTION. NA568 Mobile Robotics: Methods & Algorithms

Transcription

1 L03. PROBABILITY REVIEW II COVARIANCE PROJECTION NA568 Mobile Robotics: Methods & Algorithms

2 Today s Agenda State Representation and Uncertainty Multivariate Gaussian Covariance Projection

3 Probabilistic State Estimation Uncertain observations Sensor noise & non-idealities Uncertain beliefs Derived from sensor observations Approximate algorithms Probabilistic State Estimation Identify the quantities (state variables) we care about. e.g., study time versus exam grade Determine probability for every possible simultaneous assignment

4 Representing State Represent everything we need to know in terms of a vector of quantities State vector Usually continuous-valued in this course The meaning of the variables is up to us e.g., index 7 is the temperature in Seattle. Bookkeeping work for us.

5 Representing Uncertainty In principle, distribution of unknown quantities can be arbitrary exam grade study time

6 Correlations Estimates of variables tend to become correlated over time Observation: Study time is 4 hours Belief about study time and exam grade are affected Distribution of exam grade depends on study time: two are correlated We ll look at correlations closer later today The data does not necessarily imply any causal relationship. exam grade study time

7 Probability Basics Discrete Probability P(x) = Probability of event occurring Continuous Probability p(x) = Probability density at x

8 Probability Basics: Expectation Weighted average according to probability Basic properties of expectation

9 Joint Expectation Uncorrelated: Independence Uncorrelated Uncorrelated X Independence e.g. Conditional Expectation: implies neither independence nor uncorrelatedness e.g.

10 Variance & Covariance Average squared deviation from the mean. (Auto) covariance Scalar: Vector: (Cross) covariance Scalar: Vector:

11 Expectation Exercise We know that: Suppose we measure a bunch of samples of x. We compute the first and second moments of x, i.e., How do we compute using only these moments and the number of samples?

12 Projecting Covariances Suppose I know How do we handle??? (Algebra)

13 Properties of the Covariance Matrix Symmetric why? Positive (semi) definite why? Inverse is also positive (semi) definite Proof: see next slide Determinant Volume of uncertainty (Product of the Eigenvalues)

14 Positive (Semi) Definite Properties 1. If A 0 and B 0 then A+B 0 2. If either A or B is positive definite, then so is A+B; this follows from If A>0, then A -1 >0 4. If A 0, then F T AF 0 for any (not necessarily square) matrix F for which F T AF is defined. 5. If A>0 and F invertible, then F T AF >0. This follows from 3 and 4.

15 Correlation Coefficient The correlation coefficient is defined as: Covariance matrix in terms of correlation coefficients

16 Gaussian (Covariance Form) In this class, we ll (mostly) focus on Gaussian distributions For both observations and our beliefs Characterized by mean & covariance

17 Gaussian (Covariance Form) All that gunk out front is just for normalization. Your mental model:

18 Gaussian (Information Form) 18 An alternative parameterization of the Gaussian distribution

19 Gaussian (Information Form) 19 Information matrix and vector Encodes a graphical model Markov Random Field (a.k.a. Markov Network) Sparsity => Missing Edges => Available Conditional Independence

20 20 Gaussian Covariance & Information Parameterizations: A Dual Relationship Covariance Form Information Form Marginalization (sub-block) (Schur complement) Conditioning (Schur complement) (sub-block)

21 Visualizing Gaussians Recall our pdf: Find contours of constant probability is chi-squared distributed with n-degrees of freedom k (i.e., its square root) is known as the Mahalanobis distance Expand these terms, we end up with quadratic curve An ellipse

22 Visualizing Gaussians Number of particles within each ellipse can be computed based on properties of Gaussian distributions Actually related to Sigma 1D 2D

23 Visualizing Gaussians Number of particles within each ellipse can be computed based on properties of Gaussian distributions Actually related to Sigma 1D 2D 1 chi2cdf(1,1) chi2cdf(1,2) 2 chi2cdf(4,1) chi2cdf(4,2) 3 chi2cdf(9,1) chi2cdf(9,2)

24 2-DOF Gaussian Correlation

25 2-DOF Gaussian Correlation rho = 0 Sigma = V = D =

26 2-DOF Gaussian Correlation rho = Sigma = V = D =

27 2-DOF Gaussian Correlation rho = Sigma = V = D =

28 2-DOF Gaussian Correlation rho = 1 Sigma = V = D =

29 Why use Gaussians? Convenience Compact representation Linear operations on Gaussians produce new Gaussians Central Limit Theorem: Distribution of the sum (or average) of N independent and identically distributed (IID) random variables approaches a normal distribution. Only minor restrictions on the distribution of the individual random variables

30 Implications of CLT We often estimate the state of something using many observations Measuring the gravity on the moon by dropping a weight and timing the result Even if the distribution of each observation is non-gaussian, their average will tend towards one. CLT Visualization Demo

31 Estimating Uncertainty Where do uncertainty estimates come from? Empirically measure uncertainty Manufacturer data sheets Educated guesses Validate with error We ll come back to this

32 Odometry Example

33 Odometry Example How to convert left/right ticks to a change in position?

34 Odometry Example Sensors observe: Counts on left and right wheels No noise in those counts, however, there s slippage. Model distance as: Noise are iid Gaussian:

35 Odometry Example What is the uncertainty of? First, what s the uncertainty of d A w z But what s???

36 Odometry Example z But what s??? Remember, we said and were error-free, (iid)

37 Odometry Example We are half-way there now! Does this make intuitive sense? Answer is 2 2? No alphas?

38 Odometry Example Where are we going again? Trying to compute uncertainty of odometry measurements We know these in terms of : We ve gone from to Now, we need to go from to

39 Odometry Example Write x in terms of d x B d z

40 Odometry Example We re done! Cross-correlations happen to cancel out This does not happen in general!

41 Could do this all in one step d A w x B d

42 Sampling from Gaussians Sample from Gaussian y where Generate Gaussian noise w with return Sample from Gaussian Factor If PD, Cholesky gives a unique lower triangular L If PSD, Eigendecomposition gives a (non-unique) factorization L=VD ½ Generate Gaussian noise w with return

43 Projecting covariances (non-linear case) Again, suppose: Approach: approximate f(x) with Taylor expansion What point should we approximate f(x) around?

44 Projecting covariances (non-linear case) First-order Taylor expansion Let s review 1D case y x 0 x

45 Projecting covariances (non-linear case) Generalized case: Jacobian

46 Projecting covariances (non-linear case) A b Non-linear case is reduced to linear case via first-order Taylor approximation. Expansion point x 0 is typically taken as the mean ¹ x. What do we lose by dropping higher order terms? (see PS1 Task 2)

47 Projecting covariances (non-linear case) Summary: In non-linear case, the projected covariance depends only on the Jacobian and the covariance of the input variables! We ll be computing lots of Jacobians in this course! Lazy? Matlab/Maple can help (or sometimes totally obfuscate the answer)

48 Next Lecture Rigid body transformations and pose-chains Read: Smith, Self, & Cheesemen Eustice A1